Problem: $\vec u = -10\hat i +(-9)\hat j$ Find the direction angle of $\vec u$. Enter your answer as an angle in degrees between $0 ^\circ$ and $360^\circ$ rounded to the nearest hundredth. $\theta =$
Solution: What is a direction angle? The direction angle, $\theta$, of $\vec{u}$ is the angle between the positive $x$ -axis and $\vec{u}$. $y$ $x$ $(-10, -9)$ $\vec u$ $\theta$ Using the inverse tangent function Let's think about the components of $\vec u$ and use the inverse tangent function, $\tan^{-1}$ (also sometimes called arctangent and written as $\arctan$ or $\text{atan}$ ) to find $\theta$. $y$ $x$ $(-10, -9)$ $\vec u$ $\theta$ $-9}$ $-10}$ $\theta = \text{tan}^{-1} \left ( \dfrac{\text{Vertical component}}{\text{Horizontal component}} \right) ~~~$ $\theta=\text{tan}^{-1}\left(\dfrac{-9}{-10}\right)$ $\theta\approx{41.99^\circ} {~~~~~~~\text{WARNING: This is not the correct answer.}}$ Something isn't right here! The angle ${41.99^\circ}$ is in the first quadrant, but $\vec u$ is in the third quadrant. Shifting by $180^\circ$ because $\vec u$ is in the third quadrant Key idea: The inverse tangent function only outputs values between $-90^\circ$ and $90^\circ$, which is why our calculator didn't give us the answer we were looking for. Since $\vec u$ is in the third quadrant, we must shift the result by $180^\circ$. This makes sense if we think about it visually: $y$ $x$ $(-10, -9)$ $\vec u$ ${221.99^\circ}$ $~~~~~41.99^\circ$ $\theta \approx {41.99^\circ} + 180^\circ$ $\phantom{\theta} \approx 221.99^\circ$ The answer $\theta \approx 221.99^\circ$